1991 AIME Problems/Problem 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
Let , and denote the number of red and blue socks, respectively. Also, let . The probability that when two socks are drawn randomly, without replacement, both are red or both are blue is given by
Solving the resulting quadratic equation , for in terms of , one obtains that
Now, since and are positive integers, it must be the case that , with . Hence, would correspond to the general solution. For the present case, and so one easily finds that is the largest possible integer satisfying the problem conditions.
In summary, the solution is that the maximum number of red socks is .
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